Trapezoidal Rule of Integration

Trapezoidal Rule of Integration - Chapter 07.02 Trapezoidal...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 07.02 Trapezoidal Rule of Integration After reading this chapter, you should be able to: 1. derive the trapezoidal rule of integration, 2. use the trapezoidal rule of integration to solve problems, 3. derive the multiple-segment trapezoidal rule of integration, 4. use the multiple-segment trapezoidal rule of integration to solve problems, and 5. derive the formula for the true error in the multiple-segment trapezoidal rule of integration. What is integration? Integration is the process of measuring the area under a function plotted on a graph. Why would we want to integrate a function? Among the most common examples are finding the velocity of a body from an acceleration function, and displacement of a body from a velocity function. Throughout many engineering fields, there are (what sometimes seems like) countless applications for integral calculus. You can read about some of these applications in Chapters 07.00A-07.00G. Sometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. For this reason, a wide variety of numerical methods has been developed to simplify the integral. Here, we will discuss the trapezoidal rule of approximating integrals of the form ( 29 = b a dx x f I where ) ( x f is called the integrand, = a lower limit of integration = b upper limit of integration What is the trapezoidal rule? The trapezoidal rule is based on the Newton-Cotes formula that if one approximates the integrand by an th n order polynomial, then the integral of the function is approximated by 07.02.1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
07.02.2 Chapter 07.02 the integral of that th n order polynomial. Integrating polynomials is simple and is based on the calculus formula. Figure 1 Integration of a function 1 , 1 1 1 - + - = + + n n a b dx x n n b a n (1) So if we want to approximate the integral = b a dx x f I ) ( (2) to find the value of the above integral, one assumes ) ( ) ( x f x f n (3) where n n n n n x a x a x a a x f + + + + = - - 1 1 1 0 ... ) ( . (4) where ) ( x f n is a th n order polynomial. The trapezoidal rule assumes 1 = n , that is, approximating the integral by a linear polynomial (straight line), b a b a dx x f dx x f ) ( ) ( 1 Derivation of the Trapezoidal Rule Method 1: Derived from Calculus b a b a dx x f dx x f ) ( ) ( 1 + = b a dx x a a ) ( 1 0 - + - = 2 ) ( 2 2 1 0 a b a a b a (5)
Image of page 2
Trapezoidal Rule 07.02.3 But what is 0 a and 1 a ? Now if one chooses, )) ( , ( a f a and )) ( , ( b f b as the two points to approximate ) ( x f by a straight line from a to b , a a a a f a f 1 0 1 ) ( ) ( + = = (6) b a a b f b f 1 0 1 ) ( ) ( + = = (7) Solving the above two equations for 1 a and 0 a , a b a f b f a - - = ) ( ) ( 1 a b a b f b a f a - - = ) ( ) ( 0 (8a) Hence from Equation (5), 2 ) ( ) ( ) ( ) ( ) ( ) ( 2 2 a b a b a f b f a b a b a b f b a f dx x f b a - - - + - - - (8b) + - = 2 ) ( ) ( ) ( b f a f a b (9) Method 2: Also Derived from Calculus ) ( 1 x f can also be approximated by using Newton’s divided difference polynomial as ) ( ) ( ) ( ) ( ) ( 1 a x a b a f b f a f x f - - - + = (10) Hence b a b a dx x f dx x f ) ( ) ( 1 - - - + = b a dx a x a b a f b f a f ) ( ) ( ) ( ) ( b a ax x a b a f b f x a f
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern